Problems for 2015 AIM Workshop on Dynamical Algebraic Combinatorics
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A Combinatorial Analysis of Finite Boolean Algebras
A Combinatorial Analysis of Finite Boolean Algebras Kevin Halasz [email protected] May 1, 2013 Copyright c Kevin Halasz. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.3 or any later version published by the Free Software Foundation; with no Invariant Sections, no Front-Cover Texts, and no Back-Cover Texts. A copy of the license can be found at http://www.gnu.org/copyleft/fdl.html. 1 Contents 1 Introduction 3 2 Basic Concepts 3 2.1 Chains . .3 2.2 Antichains . .6 3 Dilworth's Chain Decomposition Theorem 6 4 Boolean Algebras 8 5 Sperner's Theorem 9 5.1 The Sperner Property . .9 5.2 Sperner's Theorem . 10 6 Extensions 12 6.1 Maximally Sized Antichains . 12 6.2 The Erdos-Ko-Rado Theorem . 13 7 Conclusion 14 2 1 Introduction Boolean algebras serve an important purpose in the study of algebraic systems, providing algebraic structure to the notions of order, inequality, and inclusion. The algebraist is always trying to understand some structured set using symbol manipulation. Boolean algebras are then used to study the relationships that hold between such algebraic structures while still using basic techniques of symbol manipulation. In this paper we will take a step back from the standard algebraic practices, and analyze these fascinating algebraic structures from a different point of view. Using combinatorial tools, we will provide an in-depth analysis of the structure of finite Boolean algebras. We will start by introducing several ways of analyzing poset substructure from a com- binatorial point of view. -
The Number of Graded Partially Ordered Sets
JOURNAL OF COMBINATORIAL THEORY 6, 12-19 (1969) The Number of Graded Partially Ordered Sets DAVID A. KLARNER* McMaster University, Hamilton, Ontario, Canada Communicated by N. G. de Bruijn ABSTRACT We find an explicit formula for the number of graded partially ordered sets of rank h that can be defined on a set containing n elements. Also, we find the number of graded partially ordered sets of length h, and having a greatest and least element that can be defined on a set containing n elements. The first result provides a lower bound for G*(n), the number of posets that can be defined on an n-set; the second result provides an upper bound for the number of lattices satisfying the Jordan-Dedekind chain condition that can be defined on an n-set. INTRODUCTION The terminology we will use in connection with partially ordered sets is defined in detail in the first few pages of Birkhoff [1]; however, we need a few concepts not defined there. A rank function g maps the elements of a poset P into the chain of integers such that (i) x > y implies g(x) > g(y), and (ii) g(x) = g(y) + 1 if x covers y. A poset P is graded if at least one rank function can be defined on it; Birkhoff calls the pair (P, g) a graded poset, where g is a particular rank function defined on a poset P, so our definition differs from his. A path in a poset P is a sequence (xl ,..., x~) such that x~ covers or is covered by xi+l, for i = 1 ... -
When Algebraic Entropy Vanishes Jim Propp (U. Wisconsin) Tufts
When algebraic entropy vanishes Jim Propp (U. Wisconsin) Tufts University October 31, 2003 1 I. Overview P robabilistic DYNAMICS ↑ COMBINATORICS ↑ Algebraic DYNAMICS 2 II. Rational maps Projective space: CPn = (Cn+1 \ (0, 0,..., 0)) / ∼, where u ∼ v iff v = cu for some c =6 0. We write the equivalence class of (x1, x2, . , xn+1) n in CP as (x1 : x2 : ... : xn+1). The standard imbedding of affine n-space into projective n-space is (x1, x2, . , xn) 7→ (x1 : x2 : ... : xn : 1). The “inverse map” is (x : ... : x : x ) 7→ ( x1 ,..., xn ). 1 n n+1 xn+1 xn+1 3 Geometrical version: CPn is the set of lines through the origin in (n + 1)-space. n The point (a1 : a2 : ... : an+1) in CP cor- responds to the line a1x1 = a2x2 = ··· = n+1 an+1xn+1 in C . The intersection of this line with the hyperplane xn+1 = 1 is the point a a a ( 1 , 2 ,..., n , 1) an+1 an+1 an+1 (as long as an+1 =6 0). We identity affine n-space with the hyperplane xn+1 = 1. Affine n-space is a Zariski-dense subset of pro- jective n-space. 4 A rational map is a function from (a Zariski-dense subset of) CPn to CPm given by m rational functions of the affine coordinate variables, or, the associated function from a Zariski-dense sub- set of Cn to CPm (e.g., the “function” x 7→ 1/x on affine 1-space, associated with the function (x : y) 7→ (y : x) on projective 1-space). -
Arxiv:1508.05446V2 [Math.CO] 27 Sep 2018 02,5B5 16E10
CELL COMPLEXES, POSET TOPOLOGY AND THE REPRESENTATION THEORY OF ALGEBRAS ARISING IN ALGEBRAIC COMBINATORICS AND DISCRETE GEOMETRY STUART MARGOLIS, FRANCO SALIOLA, AND BENJAMIN STEINBERG Abstract. In recent years it has been noted that a number of combi- natorial structures such as real and complex hyperplane arrangements, interval greedoids, matroids and oriented matroids have the structure of a finite monoid called a left regular band. Random walks on the monoid model a number of interesting Markov chains such as the Tsetlin library and riffle shuffle. The representation theory of left regular bands then comes into play and has had a major influence on both the combinatorics and the probability theory associated to such structures. In a recent pa- per, the authors established a close connection between algebraic and combinatorial invariants of a left regular band by showing that certain homological invariants of the algebra of a left regular band coincide with the cohomology of order complexes of posets naturally associated to the left regular band. The purpose of the present monograph is to further develop and deepen the connection between left regular bands and poset topology. This allows us to compute finite projective resolutions of all simple mod- ules of unital left regular band algebras over fields and much more. In the process, we are led to define the class of CW left regular bands as the class of left regular bands whose associated posets are the face posets of regular CW complexes. Most of the examples that have arisen in the literature belong to this class. A new and important class of ex- amples is a left regular band structure on the face poset of a CAT(0) cube complex. -
From the Editor
Department of Mathematics University of Wisconsin From the Editor. This year’s biggest news is the awarding of the National Medal of Science to Carl de Boor, professor emeritus of mathematics and computer science. Accompanied by his family, Professor de Boor received the medal at a White House ceremony on March 14, 2005. Carl was one of 14 new National Medal of Science Lau- reates, the only one in the category of mathematics. The award, administered by the National Science Foundation originated with the 1959 Congress. It honors individuals for pioneering research that has led to a better under- standing of the world, as well as to innovations and tech- nologies that give the USA a global economic edge. Carl is an authority on the theory and application of splines, which play a central role in, among others, computer- aided design and manufacturing, applications of com- puter graphics, and signal and image processing. The new dean of the College of Letters and Science, Gary Sandefur, said “Carl de Boor’s selection for the na- tion’s highest scientific award reflects the significance of his work and the tradition of excellence among our mathematics and computer science faculty.” We in the Department of Mathematics are extremely proud of Carl de Boor. Carl retired from the University in 2003 as Steen- bock Professor of Mathematical Sciences and now lives in Washington State, although he keeps a small condo- minium in Madison. Last year’s newsletter contains in- Carl de Boor formation about Carl’s distinguished career and a 65th birthday conference held in his honor in Germany. -
Friday 1/18/08
Friday 1/18/08 Posets Definition: A partially ordered set or poset is a set P equipped with a relation ≤ that is reflexive, antisymmetric, and transitive. That is, for all x; y; z 2 P : (1) x ≤ x (reflexivity). (2) If x ≤ y amd y ≤ x, then x = y (antisymmetry). (3) If x ≤ y and y ≤ z, then x ≤ z (transitivity). We'll usually assume that P is finite. Example 1 (Boolean algebras). Let [n] = f1; 2; : : : ; ng (a standard piece of notation in combinatorics) and let Bn be the power set of [n]. We can partially order Bn by writing S ≤ T if S ⊆ T . 123 123 12 12 13 23 12 13 23 1 2 1 2 3 1 2 3 The first two pictures are Hasse diagrams. They don't include all relations, just the covering relations, which are enough to generate all the relations in the poset. (As you can see on the right, including all the relations would make the diagram unnecessarily complicated.) Definitions: Let P be a poset and x; y 2 P . • x is covered by y, written x l y, if x < y and there exists no z such that x < z < y. • The interval from x to y is [x; y] := fz 2 P j x ≤ z ≤ yg: (This is nonempty if and only if x ≤ y, and it is a singleton set if and only if x = y.) The Boolean algebra Bn has a unique minimum element (namely ;) and a unique maximum element (namely [n]). Not every poset has to have such elements, but if a poset does, we'll call them 0^ and 1^ respectively. -
Local Characteristics, Entropy and Limit Theorems for Spanning Trees and Domino Tilings Via Transfer-Impedances
LOCAL CHARACTERISTICS, ENTROPY AND LIMIT THEOREMS FOR SPANNING TREES AND DOMINO TILINGS VIA TRANSFER-IMPEDANCES Running Head: LOCAL BEHAVIOR OF SPANNING TREES Robert Burton 1 Robin Pemantle 2 3 Oregon State University ABSTRACT: Let G be a finite graph or an infinite graph on which ZZd acts with finite fundamental domain. If G is finite, let T be a random spanning tree chosen uniformly from all spanning trees of G; if G is infinite, methods from [Pem] show that this still makes sense, producing a random essential spanning forest of G. A method for calculating local characteristics (i.e. finite-dimensional marginals) of T from the transfer-impedance matrix is presented. This differs from the classical matrix-tree theorem in that only small pieces of the matrix (n-dimensional minors) are needed to compute small (n-dimensional) marginals. Calculation of the matrix entries relies on the calculation of the Green's function for G, which is not a local calculation. However, it is shown how the calculation of the Green's function may be reduced to a finite computation in the case when G is an infinite graph admitting a Zd-action with finite quotient. The same computation also gives the entropy of the law of T. These results are applied to the problem of tiling certain lattices by dominos { the so-called dimer problem. Another application of these results is to prove modified versions of conjectures of Aldous [Al2] on the limiting distribution of degrees of a vertex and on the local structure near a vertex of a uniform random spanning tree in a lattice whose dimension is going to infinity. -
Lecture Notes on Algebraic Combinatorics Jeremy L. Martin
Lecture Notes on Algebraic Combinatorics Jeremy L. Martin [email protected] December 3, 2012 Copyright c 2012 by Jeremy L. Martin. These notes are licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 Unported License. 2 Foreword The starting point for these lecture notes was my notes from Vic Reiner's Algebraic Combinatorics course at the University of Minnesota in Fall 2003. I currently use them for graduate courses at the University of Kansas. They will always be a work in progress. Please use them and share them freely for any research purpose. I have added and subtracted some material from Vic's course to suit my tastes, but any mistakes are my own; if you find one, please contact me at [email protected] so I can fix it. Thanks to those who have suggested additions and pointed out errors, including but not limited to: Logan Godkin, Alex Lazar, Nick Packauskas, Billy Sanders, Tony Se. 1. Posets and Lattices 1.1. Posets. Definition 1.1. A partially ordered set or poset is a set P equipped with a relation ≤ that is reflexive, antisymmetric, and transitive. That is, for all x; y; z 2 P : (1) x ≤ x (reflexivity). (2) If x ≤ y and y ≤ x, then x = y (antisymmetry). (3) If x ≤ y and y ≤ z, then x ≤ z (transitivity). We'll usually assume that P is finite. Example 1.2 (Boolean algebras). Let [n] = f1; 2; : : : ; ng (a standard piece of notation in combinatorics) and let Bn be the power set of [n]. We can partially order Bn by writing S ≤ T if S ⊆ T . -
On the Rank Function of a Differential Poset Arxiv:1111.4371V2 [Math.CO] 28 Apr 2012
On the rank function of a differential poset Richard P. Stanley Fabrizio Zanello Department of Mathematics Department of Mathematical Sciences MIT Michigan Tech Cambridge, MA 02139-4307 Houghton, MI 49931-1295 [email protected] [email protected] Submitted: November 19, 2011; Accepted: April 15, 2012; Published: XX Mathematics Subject Classifications: Primary: 06A07; Secondary: 06A11, 51E15, 05C05. Abstract We study r-differential posets, a class of combinatorial objects introduced in 1988 by the first author, which gathers together a number of remarkable combinatorial and algebraic properties, and generalizes important examples of ranked posets, in- cluding the Young lattice. We first provide a simple bijection relating differential posets to a certain class of hypergraphs, including all finite projective planes, which are shown to be naturally embedded in the initial ranks of some differential poset. As a byproduct, we prove the existence, if and only if r ≥ 6, of r-differential posets nonisomorphic in any two consecutive ranks but having the same rank function. We also show that the Interval Property, conjectured by the second author and collabo- rators for several sequences of interest in combinatorics and combinatorial algebra, in general fails for differential posets. In the second part, we prove that the rank function pnpof any arbitrary r-differential poset has nonpolynomial growth; namely, a 2 rn pn n e ; a bound very close to the Hardy-Ramanujan asymptotic formula that holds in the special case of Young's lattice. We conclude by posing several open questions. Keywords: Partially ordered set, Differential poset, Rank function, Young lattice, Young-Fibonacci lattice, Hasse diagram, Hasse walk, Hypergraph, Finite projective plane, Steiner system, Interval conjecture. -
A FIRST COURSE in PROBABILITY This Page Intentionally Left Blank a FIRST COURSE in PROBABILITY
A FIRST COURSE IN PROBABILITY This page intentionally left blank A FIRST COURSE IN PROBABILITY Eighth Edition Sheldon Ross University of Southern California Upper Saddle River, New Jersey 07458 Library of Congress Cataloging-in-Publication Data Ross, Sheldon M. A first course in probability / Sheldon Ross. — 8th ed. p. cm. Includes bibliographical references and index. ISBN-13: 978-0-13-603313-4 ISBN-10: 0-13-603313-X 1. Probabilities—Textbooks. I. Title. QA273.R83 2010 519.2—dc22 2008033720 Editor in Chief, Mathematics and Statistics: Deirdre Lynch Senior Project Editor: Rachel S. Reeve Assistant Editor: Christina Lepre Editorial Assistant: Dana Jones Project Manager: Robert S. Merenoff Associate Managing Editor: Bayani Mendoza de Leon Senior Managing Editor: Linda Mihatov Behrens Senior Operations Supervisor: Diane Peirano Marketing Assistant: Kathleen DeChavez Creative Director: Jayne Conte Art Director/Designer: Bruce Kenselaar AV Project Manager: Thomas Benfatti Compositor: Integra Software Services Pvt. Ltd, Pondicherry, India Cover Image Credit: Getty Images, Inc. © 2010, 2006, 2002, 1998, 1994, 1988, 1984, 1976 by Pearson Education, Inc., Pearson Prentice Hall Pearson Education, Inc. Upper Saddle River, NJ 07458 All rights reserved. No part of this book may be reproduced, in any form or by any means, without permission in writing from the publisher. Pearson Prentice Hall™ is a trademark of Pearson Education, Inc. Printed in the United States of America 10987654321 ISBN-13: 978-0-13-603313-4 ISBN-10: 0-13-603313-X Pearson Education, Ltd., London Pearson Education Australia PTY. Limited, Sydney Pearson Education Singapore, Pte. Ltd Pearson Education North Asia Ltd, Hong Kong Pearson Education Canada, Ltd., Toronto Pearson Educacion´ de Mexico, S.A. -
Bier Spheres and Posets
Bier spheres and posets Anders Bjorner¨ ∗ Andreas Paffenholz∗∗ Dept. Mathematics Inst. Mathematics, MA 6-2 KTH Stockholm TU Berlin S-10044 Stockholm, Sweden D-10623 Berlin, Germany [email protected] [email protected] Jonas Sjostrand¨ Gunter¨ M. Ziegler∗∗∗ Dept. Mathematics Inst. Mathematics, MA 6-2 KTH Stockholm TU Berlin S-10044 Stockholm, Sweden D-10623 Berlin, Germany [email protected] [email protected] April 9, 2004 Dedicated to Louis J. Billera on occasion of his 60th birthday Abstract In 1992 Thomas Bier presented a strikingly simple method to produce a huge number of simplicial (n − 2)-spheres on 2n vertices as deleted joins of a simplicial complex on n vertices with its combinatorial Alexander dual. Here we interpret his construction as giving the poset of all the intervals in a boolean algebra that “cut across an ideal.” Thus we arrive at a substantial generalization of Bier’s construction: the Bier posets Bier(P,I) of an arbitrary bounded poset P of finite length. In the case of face posets of PL spheres this yields arXiv:math/0311356v2 [math.CO] 12 Apr 2004 cellular “generalized Bier spheres.” In the case of Eulerian or Cohen-Macaulay posets P we show that the Bier posets Bier(P,I) inherit these properties. In the boolean case originally considered by Bier, we show that all the spheres produced by his construction are shellable, which yields “many shellable spheres,” most of which lack convex realization. Finally, we present simple explicit formulas for the g-vectors of these simplicial spheres and verify that they satisfy a strong form of the g-conjecture for spheres. -
Perfect Matchings for the Three-Term Gale-Robinson Sequences Mireille Bousquet-Mélou, James Propp, Julian West
Perfect matchings for the three-term Gale-Robinson sequences Mireille Bousquet-Mélou, James Propp, Julian West To cite this version: Mireille Bousquet-Mélou, James Propp, Julian West. Perfect matchings for the three-term Gale- Robinson sequences. The Electronic Journal of Combinatorics, Open Journal Systems, 2009, 16 (1), paper R125. hal-00396223 HAL Id: hal-00396223 https://hal.archives-ouvertes.fr/hal-00396223 Submitted on 17 Jun 2009 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. PERFECT MATCHINGS FOR THE THREE-TERM GALE-ROBINSON SEQUENCES MIREILLE BOUSQUET-MELOU,´ JAMES PROPP, AND JULIAN WEST Abstract. In 1991, David Gale and Raphael Robinson, building on explorations carried out by Michael Somos in the 1980s, introduced a three-parameter family of rational recurrence relations, each of which (with suitable initial conditions) appeared to give rise to a sequence of integers, even though a priori the recurrence might produce non-integral rational numbers. Throughout the ’90s, proofs of integrality were known only for individual special cases. In the early ’00s, Sergey Fomin and Andrei Zelevinsky proved Gale and Robinson’s integrality conjecture. They actually proved much more, and in particular, that certain bivariate ratio- nal functions that generalize Gale-Robinson numbers are actually polynomials with integer coefficients.